Electrical Conduction
in Solids
Physics 127
Introduction to Modern
Physics Laboratory
Department of Physics
HKUST
Revised: 23 January 2009
Phys127 Introduction to Modern Physics Lab
Electrical conduction in solids
Background
The investigation of the temperature dependence of electrical resistance provides an interesting
insight into the nature of electrical conduction processes in solids. Especially striking are the
differences between metallic conductors and semiconductors. By studying a noble metal, you
will be able to verify the basic prediction of the simplest Drude model of conduction in solids.
Examination of conduction in a semiconductor, on the other hand, will give you insight into the
concepts of the energy gap, Fermi-Dirac Statistics, and Boltzmann factor.
Conduction in Metals: The Drude Model
In a conductor, the electrons in the outermost atomic shells which are most weakly bound to the
atoms are free to move around the solid. Occasionally, these conduction electrons scatter off of
the atomic cores as they move through the solid. The random motion of the conduction electrons
leads to no net flow of electrons in any one direction, and therefore there is no net current. When
an electric field is applied across the conductor, the conduction electrons drift slowly under the
influence of the electrostatic force, F = e E. However, their acceleration is not constant because
of occasional collisions with atoms in their paths. These collisions are effectively a drag force
which is opposed to the electrostatic force. Under the influence of the drag force, the electrons
eventually move through the solid with a constant drift velocity vd given by
vd =
eEt
m
where t is some characteristic relaxation time between collisions which depends upon many
things including the vibrational motion of the atoms and m is the mass of electrons.
The current passing through a wire of cross-sectional area A is equal to the charge passing
through the cross-sectional surface, dq, per unit time, dt. The charge dq is equal to the total
number of conduction electrons in a volume of wire of length vddt and cross-sectional area A
times the electron charge, e. Therefore,
dq = neAv d ⋅ dt
where n is the number density per unit volume of conduction electrons. The current I is then
I=
dq
= neAv d
dt
The potential difference V across a length L of wire is given by
V = − ∫ Eidl = EL
Substituting I and V into ohm's law we get
R=
V
EL
=
I neAvd
⎛ m ⎞ L
⇒ R = ⎜ 2 ⎟⋅
⎝ ne t ⎠ A
The term in brackets is called the resistivity, r, which is the inverse of the conductivity, s. In this
experiment, you will examine the resistance, which depends upon temperature essentially
through the relaxation time, t, due to atomic vibrations. The temperature dependence of the
resistance can be approximated reasonably well by the expression
R θ = R 0 (1 + βθ )
(1)
where R0 is the resistance at 0°C, θ is the temperature in °C, R θ is the resistance at temperature
θ, and β is the temperature coefficient of the resistor (resistance change per degree).
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Phys127 Introduction to Modern Physics Lab
Electrical conduction in solids
Conduction in Semiconductors: The Energy Gap, Boltzmann Factor, and Fermi-Dirac Statistics
It is known that electrons can occupy only discrete energy levels in atoms, which are reasonably
described by simple theories such as Bohr's model. Similarly, the valence electrons, i.e. the
electrons which are most weakly bound to the atoms, also occupy well-defined energy bands in
the solid formed by a collection of atoms. Each energy band can contain many electrons (as
shown in Figure 1). If an energy band is only partially filled with electrons (Fig. 1b), the material
will conduct electrons in the simple way described by the Drude model above. On the other
hand, if there are no partially filled energy bands (Fig. 1a), i.e. only fully filled and fully empty
bands, the material is called an insulator or semiconductor. The unoccupied band, in this case,
called the conduction band and the occupied band called the valence band. The energy
difference between the valence and conduction bands called the energy gap Eg.
Figure 1 Energy E versus density of levels g(E) 1
(a) In an insulator or semiconductor, there is a regoin of forbidden energies seperating the highest occupied and
lowest unoccupied levels.
(b) In a metal, the boundary occurs in a region of allowed levels
Electrical conduction in a semiconductor can only occur when electrons from the valence band
jump across the energy gap into the conduction band. The number of electrons which have made
this jump can be calculated using the Fermi-Dirac distribution. This distribution states that the
occupancy of an energy level, f(E), is given by:
f(E) =
1
1+e
(E - E F )
k BT
where E F is the Fermi energy which is halfway between the valence and conduction bands
(NOTE: actually, this is not precisely true for all semiconductors but it is a reasonably good
point to start your understanding of semiconductor materials), kB is the Boltzmann constant, and
T is the temperature in degrees Kelvin. From this distribution, it can be shown 2 that the number
of electrons which have jumped across the gap to the conduction band - called the carrier density
− E /2k T
- is approximately proportional to the factor e g B . Note, that this looks very much like the
Boltzmann factor, e − Ei /k BT . We can see from this result that the carrier density is zero at T = 0 K
and increases at higher temperature. Thus, electrical conductivity increases, or equivalently the
resistance decreases, at higher temperature which is opposite to the behaviour of metals.
1
2
Neil W Ashcroft, N. David Mermin, Solid State Physics, c1976, p.562, Figure 28.1
Neil W Ashcroft, N. David Mermin, Solid State Physics, c1976, p.572-575
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Phys127 Introduction to Modern Physics Lab
Electrical conduction in solids
Procedure
In the experiment, the resistance values of a noble-metal resistor and a semiconductor resistor as
a function of the temperature will be measured.
Introduction to equipment and components:
1. Sample resistors – Two resistors, say noble-metal and semiconductor, will be investigated.
They are sealed in a glass tube and mounted on a carrier plate with two 4mm sockets for
resistance measurement. The temperature range for the resistor is:
Noble-metal resistor
-80 °C to 400 °C
Semiconductor resistor
-80 °C to 200 °C
2. Oven with temperature controller – A ceramic oven with cylindrical heating chamber is used
in heating the resistor. The temperature controller will help to set the oven to a desired
setpoint by a control loop feedback mechanism called PID (which stands for proportional–
integral–derivative).
Resistance (R) versus temperature (T) measurement
1. Set up the equipment as shown in Figure 2.
NOTE: To save time, the oven may be pre-heated for a while, please beware of the HOT
oven and sample resistor during the experiment.
2. Start with the semiconductor resistor as the sample resisitor
3. Ensure the tip of the K-type thermocouple probe is directly next to the sample resistor.
4. Set the temperature setpoint to 40 °C by following the instructions on the temperature
controller.
5. Once the temperature is stable (near the setpoint value), record and tabulate the
temperature T and resistance R of the sample resistor in your data sheet.
Figure 2 Experimental setup
6. Increase the temperature setpoint to its next value:
For semiconductor resistor, it is recommended to increase in 20 °C step,
for noble-metal resistor, it is recommended to increase in 50 °C step.
7. Wait until the temperature is stable, record the temperature and resistance.
8. Repeat the measurement until the temperature reaches the following value:
180 °C for semiconductor resistor,
370 °C. for noble-metal resistor.
NOTE: Never heat up the sample resistor above its maximum permissible value.
9. In the meantime, measure the resistance of the noble-metal resistor at room temperature and
ice-water point. Record the temperature and resistance in the data sheet.
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Phys127 Introduction to Modern Physics Lab
Electrical conduction in solids
10. Once the measurement of semiconductor is finished, withdraw the semiconductor resistor
from the oven and allow it to cool down. Cool down the oven to about 40 °C with the help
of a cooling fan and set the temperature setpoint to 20 °C.
11. When the oven is cooled down, inert the noble-metal resistor into the oven and set the
temperature setpoint to 70 °C.
12. Repeat the measurement as stated in step 5 to 8. In the meantime, cool down the
semiconductor resistor to room temperature.
13. Once the semiconductor is cooled down to room temperature, measure the resistance of
the semiconductor resistor at room temperature and ice-water point. Record the temperature
and resistance in the data sheet.
Data analysis
a. For the semiconductor resistor, plot the natural logarithm of the resistance (ln R) versus the
inverse of temperature (1/T, where T is in degree Kelvin). Fit the data with a linear function
and record the fitting result in the lab report.
b. For the noble-metal resistor, plot the resistance (R) versus the temperature (T, where T is in
°C). Fit the data with a linear function and record the fitting result in the lab report.
Discussion
1. Does the behaviour of the resistance (or conductance) as a function of temperature of both
sample resistors agree with theories?
2. From the fitting result of the semiconductor resistor, determine the energy gap Eg. Compare
the result with the accept value, Eg = 0.47 eV.
3. From the fitting result of the noble-metal resistor, determine the temperature coefficient β as
stated in Eq. (1). Compare the result with the literature value β = 0.00392 °C-1 for platinum.
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